The Boussinesq equation for the surface waves reduces to a 4th-order elliptic equation for steady moving waves. This is investigated for bifurcation in 2D by devising a finite-difference scheme and an iterative algorithm. We prove that the truncation error of the scheme is second-order in spac. Next, we develop a perturbation series with respect to the small parameter (square of the phase speed of the wave). Within 2nd order of the small parameter, we derive a hierarchy of 1D equations that are 4th-order in the radial variable and solve the ODEs. We create special approximations to handle the so-called behavioral conditions at the point of singularity. Comparison of the results obtained with the two different techniques is in excellent agreement and validated. We discover that the shape of the moving soliton decays as inverse-square of the radial distance from the center of the base, while the profile of the standing soliton decays exponentially. This means that the asymptotic behavior of the solution is not robust, a novel result. Our results are of importance both for the mathematical theory of Boussinesq solitons in multi-dimension, and for their physical applications.